{
 "nbformat": 4,
 "nbformat_minor": 2,
 "metadata": {
  "language_info": {
   "name": "python",
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "version": "3.6.8-final"
  },
  "orig_nbformat": 2,
  "file_extension": ".py",
  "mimetype": "text/x-python",
  "name": "python",
  "npconvert_exporter": "python",
  "pygments_lexer": "ipython3",
  "version": 3,
  "kernelspec": {
   "name": "python36864bit3f53e4684f024fc68f038ddcedf7a612",
   "display_name": "Python 3.6.8 64-bit"
  }
 },
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# population models"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>year</th>\n      <th>population</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>0</th>\n      <td>1790.0</td>\n      <td>3.9</td>\n    </tr>\n    <tr>\n      <th>1</th>\n      <td>1800.0</td>\n      <td>5.3</td>\n    </tr>\n    <tr>\n      <th>2</th>\n      <td>1810.0</td>\n      <td>7.2</td>\n    </tr>\n    <tr>\n      <th>3</th>\n      <td>1820.0</td>\n      <td>9.6</td>\n    </tr>\n    <tr>\n      <th>4</th>\n      <td>1830.0</td>\n      <td>12.9</td>\n    </tr>\n    <tr>\n      <th>5</th>\n      <td>1840.0</td>\n      <td>17.1</td>\n    </tr>\n    <tr>\n      <th>6</th>\n      <td>1850.0</td>\n      <td>23.2</td>\n    </tr>\n    <tr>\n      <th>7</th>\n      <td>1860.0</td>\n      <td>31.4</td>\n    </tr>\n    <tr>\n      <th>8</th>\n      <td>1870.0</td>\n      <td>38.6</td>\n    </tr>\n    <tr>\n      <th>9</th>\n      <td>1880.0</td>\n      <td>50.2</td>\n    </tr>\n    <tr>\n      <th>10</th>\n      <td>1890.0</td>\n      <td>62.9</td>\n    </tr>\n    <tr>\n      <th>11</th>\n      <td>1900.0</td>\n      <td>76.0</td>\n    </tr>\n    <tr>\n      <th>12</th>\n      <td>1910.0</td>\n      <td>92.0</td>\n    </tr>\n    <tr>\n      <th>13</th>\n      <td>1920.0</td>\n      <td>106.5</td>\n    </tr>\n    <tr>\n      <th>14</th>\n      <td>1930.0</td>\n      <td>123.2</td>\n    </tr>\n    <tr>\n      <th>15</th>\n      <td>1940.0</td>\n      <td>131.7</td>\n    </tr>\n    <tr>\n      <th>16</th>\n      <td>1950.0</td>\n      <td>150.7</td>\n    </tr>\n    <tr>\n      <th>17</th>\n      <td>1960.0</td>\n      <td>179.3</td>\n    </tr>\n    <tr>\n      <th>18</th>\n      <td>1970.0</td>\n      <td>204.0</td>\n    </tr>\n    <tr>\n      <th>19</th>\n      <td>1980.0</td>\n      <td>226.5</td>\n    </tr>\n    <tr>\n      <th>20</th>\n      <td>1990.0</td>\n      <td>251.4</td>\n    </tr>\n  </tbody>\n</table>\n</div>",
      "text/plain": "      year  population\n0   1790.0         3.9\n1   1800.0         5.3\n2   1810.0         7.2\n3   1820.0         9.6\n4   1830.0        12.9\n5   1840.0        17.1\n6   1850.0        23.2\n7   1860.0        31.4\n8   1870.0        38.6\n9   1880.0        50.2\n10  1890.0        62.9\n11  1900.0        76.0\n12  1910.0        92.0\n13  1920.0       106.5\n14  1930.0       123.2\n15  1940.0       131.7\n16  1950.0       150.7\n17  1960.0       179.3\n18  1970.0       204.0\n19  1980.0       226.5\n20  1990.0       251.4"
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      "text/plain": "<Figure size 432x288 with 1 Axes>"
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np \n",
    "import pandas as pd\n",
    "from matplotlib import pyplot as plt \n",
    "from scipy.optimize import curve_fit\n",
    "\n",
    "data = pd.DataFrame(pd.read_csv('data.csv'),dtype=float)\n",
    "plt.scatter(data.year,data.population)\n",
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "data_size = len(data)\n",
    "year = np.arange(data_size)\n",
    "\n",
    "def loss(pred,real):\n",
    "    return np.abs(pred-real)/real\n",
    "\n",
    "def percentage(x):\n",
    "    return \"%.2f%%\" % (x * 100)\n",
    "\n",
    "def fit_display(f):\n",
    "    popt, _ = curve_fit(f,year,data.population/1000)\n",
    "    print('parameters:',popt)\n",
    "    pred = f(year,*popt)*1000\n",
    "\n",
    "    result = data.copy()\n",
    "    result['prediction'] = np.round(pred,2)\n",
    "    loss_ = loss(pred,data.population)\n",
    "    result['loss'] = np.asarray(list(map(percentage,loss_)))\n",
    "    print('loss:',percentage(np.mean(loss_)))\n",
    "\n",
    "    plt.scatter(data.year,data.population,label='real data')\n",
    "    plt.plot(data.year,pred,'r-',label = 'prediction')\n",
    "    plt.legend()\n",
    "\n",
    "    return result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## model 1\n",
    "### expression\n",
    "differential equation:\n",
    "$$$\n",
    "\\frac{dP}{dt}=\\gamma \\cdot P\n",
    "$$$\n",
    "function:\n",
    "$$$\n",
    "P(t) = P_0 \\cdot e^{\\gamma t}\n",
    "$$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "parameters: [0.21685414]\nloss: 26.96%\n"
    },
    {
     "data": {
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>year</th>\n      <th>population</th>\n      <th>prediction</th>\n      <th>loss</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>0</th>\n      <td>1790.0</td>\n      <td>3.9</td>\n      <td>3.90</td>\n      <td>0.00%</td>\n    </tr>\n    <tr>\n      <th>1</th>\n      <td>1800.0</td>\n      <td>5.3</td>\n      <td>4.84</td>\n      <td>8.60%</td>\n    </tr>\n    <tr>\n      <th>2</th>\n      <td>1810.0</td>\n      <td>7.2</td>\n      <td>6.02</td>\n      <td>16.42%</td>\n    </tr>\n    <tr>\n      <th>3</th>\n      <td>1820.0</td>\n      <td>9.6</td>\n      <td>7.47</td>\n      <td>22.14%</td>\n    </tr>\n    <tr>\n      <th>4</th>\n      <td>1830.0</td>\n      <td>12.9</td>\n      <td>9.28</td>\n      <td>28.02%</td>\n    </tr>\n    <tr>\n      <th>5</th>\n      <td>1840.0</td>\n      <td>17.1</td>\n      <td>11.53</td>\n      <td>32.55%</td>\n    </tr>\n    <tr>\n      <th>6</th>\n      <td>1850.0</td>\n      <td>23.2</td>\n      <td>14.33</td>\n      <td>38.25%</td>\n    </tr>\n    <tr>\n      <th>7</th>\n      <td>1860.0</td>\n      <td>31.4</td>\n      <td>17.80</td>\n      <td>43.33%</td>\n    </tr>\n    <tr>\n      <th>8</th>\n      <td>1870.0</td>\n      <td>38.6</td>\n      <td>22.11</td>\n      <td>42.73%</td>\n    </tr>\n    <tr>\n      <th>9</th>\n      <td>1880.0</td>\n      <td>50.2</td>\n      <td>27.46</td>\n      <td>45.30%</td>\n    </tr>\n    <tr>\n      <th>10</th>\n      <td>1890.0</td>\n      <td>62.9</td>\n      <td>34.11</td>\n      <td>45.78%</td>\n    </tr>\n    <tr>\n      <th>11</th>\n      <td>1900.0</td>\n      <td>76.0</td>\n      <td>42.37</td>\n      <td>44.25%</td>\n    </tr>\n    <tr>\n      <th>12</th>\n      <td>1910.0</td>\n      <td>92.0</td>\n      <td>52.63</td>\n      <td>42.80%</td>\n    </tr>\n    <tr>\n      <th>13</th>\n      <td>1920.0</td>\n      <td>106.5</td>\n      <td>65.37</td>\n      <td>38.62%</td>\n    </tr>\n    <tr>\n      <th>14</th>\n      <td>1930.0</td>\n      <td>123.2</td>\n      <td>81.20</td>\n      <td>34.09%</td>\n    </tr>\n    <tr>\n      <th>15</th>\n      <td>1940.0</td>\n      <td>131.7</td>\n      <td>100.87</td>\n      <td>23.41%</td>\n    </tr>\n    <tr>\n      <th>16</th>\n      <td>1950.0</td>\n      <td>150.7</td>\n      <td>125.29</td>\n      <td>16.86%</td>\n    </tr>\n    <tr>\n      <th>17</th>\n      <td>1960.0</td>\n      <td>179.3</td>\n      <td>155.63</td>\n      <td>13.20%</td>\n    </tr>\n    <tr>\n      <th>18</th>\n      <td>1970.0</td>\n      <td>204.0</td>\n      <td>193.32</td>\n      <td>5.23%</td>\n    </tr>\n    <tr>\n      <th>19</th>\n      <td>1980.0</td>\n      <td>226.5</td>\n      <td>240.14</td>\n      <td>6.02%</td>\n    </tr>\n    <tr>\n      <th>20</th>\n      <td>1990.0</td>\n      <td>251.4</td>\n      <td>298.29</td>\n      <td>18.65%</td>\n    </tr>\n  </tbody>\n</table>\n</div>",
      "text/plain": "      year  population  prediction    loss\n0   1790.0         3.9        3.90   0.00%\n1   1800.0         5.3        4.84   8.60%\n2   1810.0         7.2        6.02  16.42%\n3   1820.0         9.6        7.47  22.14%\n4   1830.0        12.9        9.28  28.02%\n5   1840.0        17.1       11.53  32.55%\n6   1850.0        23.2       14.33  38.25%\n7   1860.0        31.4       17.80  43.33%\n8   1870.0        38.6       22.11  42.73%\n9   1880.0        50.2       27.46  45.30%\n10  1890.0        62.9       34.11  45.78%\n11  1900.0        76.0       42.37  44.25%\n12  1910.0        92.0       52.63  42.80%\n13  1920.0       106.5       65.37  38.62%\n14  1930.0       123.2       81.20  34.09%\n15  1940.0       131.7      100.87  23.41%\n16  1950.0       150.7      125.29  16.86%\n17  1960.0       179.3      155.63  13.20%\n18  1970.0       204.0      193.32   5.23%\n19  1980.0       226.5      240.14   6.02%\n20  1990.0       251.4      298.29  18.65%"
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      "text/plain": "<Figure size 432x288 with 1 Axes>"
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "P0 = data.population.iloc[0]/1000\n",
    "def f1(x,r):\n",
    "    return P0*np.exp(r*x)\n",
    "\n",
    "fit_display(f1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where we calculate the parameter:$\\gamma = 0.262$\n",
    "and loss here:$loss = $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "parameters: [0.23722757]\n"
    },
    {
     "data": {
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>year</th>\n      <th>population</th>\n      <th>prediction</th>\n      <th>loss</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>0</th>\n      <td>1790.0</td>\n      <td>3.9</td>\n      <td>3.90</td>\n      <td>0.00</td>\n    </tr>\n    <tr>\n      <th>1</th>\n      <td>1800.0</td>\n      <td>5.3</td>\n      <td>4.94</td>\n      <td>0.07</td>\n    </tr>\n    <tr>\n      <th>2</th>\n      <td>1810.0</td>\n      <td>7.2</td>\n      <td>6.27</td>\n      <td>0.13</td>\n    </tr>\n    <tr>\n      <th>3</th>\n      <td>1820.0</td>\n      <td>9.6</td>\n      <td>7.95</td>\n      <td>0.17</td>\n    </tr>\n    <tr>\n      <th>4</th>\n      <td>1830.0</td>\n      <td>12.9</td>\n      <td>10.07</td>\n      <td>0.22</td>\n    </tr>\n    <tr>\n      <th>5</th>\n      <td>1840.0</td>\n      <td>17.1</td>\n      <td>12.77</td>\n      <td>0.25</td>\n    </tr>\n    <tr>\n      <th>6</th>\n      <td>1850.0</td>\n      <td>23.2</td>\n      <td>16.19</td>\n      <td>0.30</td>\n    </tr>\n    <tr>\n      <th>7</th>\n      <td>1860.0</td>\n      <td>31.4</td>\n      <td>20.52</td>\n      <td>0.35</td>\n    </tr>\n    <tr>\n      <th>8</th>\n      <td>1870.0</td>\n      <td>38.6</td>\n      <td>26.02</td>\n      <td>0.33</td>\n    </tr>\n    <tr>\n      <th>9</th>\n      <td>1880.0</td>\n      <td>50.2</td>\n      <td>32.98</td>\n      <td>0.34</td>\n    </tr>\n    <tr>\n      <th>10</th>\n      <td>1890.0</td>\n      <td>62.9</td>\n      <td>41.81</td>\n      <td>0.34</td>\n    </tr>\n    <tr>\n      <th>11</th>\n      <td>1900.0</td>\n      <td>76.0</td>\n      <td>53.01</td>\n      <td>0.30</td>\n    </tr>\n    <tr>\n      <th>12</th>\n      <td>1910.0</td>\n      <td>92.0</td>\n      <td>67.20</td>\n      <td>0.27</td>\n    </tr>\n    <tr>\n      <th>13</th>\n      <td>1920.0</td>\n      <td>106.5</td>\n      <td>85.19</td>\n      <td>0.20</td>\n    </tr>\n    <tr>\n      <th>14</th>\n      <td>1930.0</td>\n      <td>123.2</td>\n      <td>108.00</td>\n      <td>0.12</td>\n    </tr>\n    <tr>\n      <th>15</th>\n      <td>1940.0</td>\n      <td>131.7</td>\n      <td>136.92</td>\n      <td>0.04</td>\n    </tr>\n    <tr>\n      <th>16</th>\n      <td>1950.0</td>\n      <td>150.7</td>\n      <td>173.58</td>\n      <td>0.15</td>\n    </tr>\n    <tr>\n      <th>17</th>\n      <td>1960.0</td>\n      <td>179.3</td>\n      <td>220.05</td>\n      <td>0.23</td>\n    </tr>\n    <tr>\n      <th>18</th>\n      <td>1970.0</td>\n      <td>204.0</td>\n      <td>278.96</td>\n      <td>0.37</td>\n    </tr>\n    <tr>\n      <th>19</th>\n      <td>1980.0</td>\n      <td>226.5</td>\n      <td>353.65</td>\n      <td>0.56</td>\n    </tr>\n    <tr>\n      <th>20</th>\n      <td>1990.0</td>\n      <td>251.4</td>\n      <td>448.33</td>\n      <td>0.78</td>\n    </tr>\n  </tbody>\n</table>\n</div>",
      "text/plain": "      year  population  prediction  loss\n0   1790.0         3.9        3.90  0.00\n1   1800.0         5.3        4.94  0.07\n2   1810.0         7.2        6.27  0.13\n3   1820.0         9.6        7.95  0.17\n4   1830.0        12.9       10.07  0.22\n5   1840.0        17.1       12.77  0.25\n6   1850.0        23.2       16.19  0.30\n7   1860.0        31.4       20.52  0.35\n8   1870.0        38.6       26.02  0.33\n9   1880.0        50.2       32.98  0.34\n10  1890.0        62.9       41.81  0.34\n11  1900.0        76.0       53.01  0.30\n12  1910.0        92.0       67.20  0.27\n13  1920.0       106.5       85.19  0.20\n14  1930.0       123.2      108.00  0.12\n15  1940.0       131.7      136.92  0.04\n16  1950.0       150.7      173.58  0.15\n17  1960.0       179.3      220.05  0.23\n18  1970.0       204.0      278.96  0.37\n19  1980.0       226.5      353.65  0.56\n20  1990.0       251.4      448.33  0.78"
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      "text/plain": "<Figure size 432x288 with 1 Axes>"
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f11(x,r):\n",
    "    return np.log(data.population.iloc[0]) + r*x\n",
    "\n",
    "popt_, _ = curve_fit(f11,year,np.log(data.population))\n",
    "print('parameters:',popt_)\n",
    "pred_ = np.exp(f11(year,*popt_))\n",
    "\n",
    "result_ = data.copy()\n",
    "result_['prediction'] = np.round(pred_,2)\n",
    "result_['loss'] = np.round(loss(pred_,data.population),2)\n",
    "\n",
    "plt.scatter(data.year,data.population,label='real data')\n",
    "plt.plot(data.year,pred_,'r-',label = 'prediction')\n",
    "result_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## model 2\n",
    "### expression\n",
    "ODE:\n",
    "$$$\n",
    "\\frac{P}{dt}=\\gamma P (1-\\frac{P}{\\kappa})\n",
    "$$$\n",
    "function:\n",
    "$$$\n",
    "P(t)=\\frac{\\kappa P_0 e^{\\gamma t}}{\\kappa+P_0 e^{\\gamma t}}\n",
    "$$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "parameters: [0.27981712 0.31195548]\nloss: 8.04%\n"
    },
    {
     "data": {
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>year</th>\n      <th>population</th>\n      <th>prediction</th>\n      <th>loss</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>0</th>\n      <td>1790.0</td>\n      <td>3.9</td>\n      <td>3.90</td>\n      <td>0.00%</td>\n    </tr>\n    <tr>\n      <th>1</th>\n      <td>1800.0</td>\n      <td>5.3</td>\n      <td>5.14</td>\n      <td>3.05%</td>\n    </tr>\n    <tr>\n      <th>2</th>\n      <td>1810.0</td>\n      <td>7.2</td>\n      <td>6.76</td>\n      <td>6.09%</td>\n    </tr>\n    <tr>\n      <th>3</th>\n      <td>1820.0</td>\n      <td>9.6</td>\n      <td>8.88</td>\n      <td>7.47%</td>\n    </tr>\n    <tr>\n      <th>4</th>\n      <td>1830.0</td>\n      <td>12.9</td>\n      <td>11.64</td>\n      <td>9.74%</td>\n    </tr>\n    <tr>\n      <th>5</th>\n      <td>1840.0</td>\n      <td>17.1</td>\n      <td>15.22</td>\n      <td>10.99%</td>\n    </tr>\n    <tr>\n      <th>6</th>\n      <td>1850.0</td>\n      <td>23.2</td>\n      <td>19.82</td>\n      <td>14.56%</td>\n    </tr>\n    <tr>\n      <th>7</th>\n      <td>1860.0</td>\n      <td>31.4</td>\n      <td>25.70</td>\n      <td>18.17%</td>\n    </tr>\n    <tr>\n      <th>8</th>\n      <td>1870.0</td>\n      <td>38.6</td>\n      <td>33.11</td>\n      <td>14.22%</td>\n    </tr>\n    <tr>\n      <th>9</th>\n      <td>1880.0</td>\n      <td>50.2</td>\n      <td>42.35</td>\n      <td>15.63%</td>\n    </tr>\n    <tr>\n      <th>10</th>\n      <td>1890.0</td>\n      <td>62.9</td>\n      <td>53.67</td>\n      <td>14.67%</td>\n    </tr>\n    <tr>\n      <th>11</th>\n      <td>1900.0</td>\n      <td>76.0</td>\n      <td>67.27</td>\n      <td>11.49%</td>\n    </tr>\n    <tr>\n      <th>12</th>\n      <td>1910.0</td>\n      <td>92.0</td>\n      <td>83.19</td>\n      <td>9.57%</td>\n    </tr>\n    <tr>\n      <th>13</th>\n      <td>1920.0</td>\n      <td>106.5</td>\n      <td>101.33</td>\n      <td>4.85%</td>\n    </tr>\n    <tr>\n      <th>14</th>\n      <td>1930.0</td>\n      <td>123.2</td>\n      <td>121.33</td>\n      <td>1.52%</td>\n    </tr>\n    <tr>\n      <th>15</th>\n      <td>1940.0</td>\n      <td>131.7</td>\n      <td>142.59</td>\n      <td>8.27%</td>\n    </tr>\n    <tr>\n      <th>16</th>\n      <td>1950.0</td>\n      <td>150.7</td>\n      <td>164.37</td>\n      <td>9.07%</td>\n    </tr>\n    <tr>\n      <th>17</th>\n      <td>1960.0</td>\n      <td>179.3</td>\n      <td>185.83</td>\n      <td>3.64%</td>\n    </tr>\n    <tr>\n      <th>18</th>\n      <td>1970.0</td>\n      <td>204.0</td>\n      <td>206.18</td>\n      <td>1.07%</td>\n    </tr>\n    <tr>\n      <th>19</th>\n      <td>1980.0</td>\n      <td>226.5</td>\n      <td>224.78</td>\n      <td>0.76%</td>\n    </tr>\n    <tr>\n      <th>20</th>\n      <td>1990.0</td>\n      <td>251.4</td>\n      <td>241.24</td>\n      <td>4.04%</td>\n    </tr>\n  </tbody>\n</table>\n</div>",
      "text/plain": "      year  population  prediction    loss\n0   1790.0         3.9        3.90   0.00%\n1   1800.0         5.3        5.14   3.05%\n2   1810.0         7.2        6.76   6.09%\n3   1820.0         9.6        8.88   7.47%\n4   1830.0        12.9       11.64   9.74%\n5   1840.0        17.1       15.22  10.99%\n6   1850.0        23.2       19.82  14.56%\n7   1860.0        31.4       25.70  18.17%\n8   1870.0        38.6       33.11  14.22%\n9   1880.0        50.2       42.35  15.63%\n10  1890.0        62.9       53.67  14.67%\n11  1900.0        76.0       67.27  11.49%\n12  1910.0        92.0       83.19   9.57%\n13  1920.0       106.5      101.33   4.85%\n14  1930.0       123.2      121.33   1.52%\n15  1940.0       131.7      142.59   8.27%\n16  1950.0       150.7      164.37   9.07%\n17  1960.0       179.3      185.83   3.64%\n18  1970.0       204.0      206.18   1.07%\n19  1980.0       226.5      224.78   0.76%\n20  1990.0       251.4      241.24   4.04%"
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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GRo28UkJiot8RRbRCE4Jz7jPybhcAOD+fc54EnixFXCIipbN6tddeULMmfPgh1K9f+DkxTiOVRaT8+fFH6NrV254zBxo39jeeKKHZTkWkfPnpJy8Z7NwJn3wCJ5/sd0RRQwlBRMqP7du9aqKNG71qojZt/I4oqighiEj5sGeP14D87bfw9tvQqZPfEUUdJQQRiX7790Pv3vDll/DGG14pQYpNCUFEotvhw946yB984E1N0bev3xFFLfUyEpHoNniwVyp4+mm44Qa/o4lqSggiEr2ef96bvfSOO7zEIKWihCAi0enNN+HPfybrvAvp/LuepAydTafhH8f0EpilpTYEEYk+n34K/fvzS+tULmp/C9t3HgAgc3s2Q6d/A6BZTEtAJQQRiS4rV0JaGpx4Ilf1fIDtub7Xxvq6yKWhhCAi0SMzE3r0gIQEeO89Vh+Iz/OwWF4XuTSUEEQkOuzYARde6D3Ong0nnJDv+sexvC5yaSghiEjk27/fW9Ng1SpvgZvAlBSDuzcjIT7uqENjfV3k0lCjsohEtsOH4frrYe5ceO2132Yx5beG42ffX03W9mwaJiYwuHszNSiXkBKCiES2IUNg0iQYNgz69z/maa2LHDqqMhKRyDVqFDz7LAwcCPff73c05Z4SgohEpmnT4O67vS6mo0aB5bdwo4SKEoKIRJ758+Gqq6BjR5g4EeLiCj9HSk0JQUQiy6pV0KsXNGkCs2Z5Yw4kLJQQRCRyZGV5A88qVYJ334U6dfyOKKaol5GIRIadO72BZ1u3enMVpaT4HVHMUUIQEf8dOAB9+njLX/7739Cund8RxSQlBBHxl3MwYADMmQPjxkG3bn5HFLPUhiAi/nr0URg/Hh55BK67zu9oYpoSgoj4Z9w4LyFcdx389a9+RxPzlBBExB8ffOBVFXXtCqNHa+BZBFBCEJHwW74c+vaF006DqVMhPu91DSS8lBBEJLwyMuDii6FGDa9HUY0afkckAeplJCLhs2MHXHSRN+bgs88gOdnviCSIEoKIhMeBA1410apV3opnrVr5HZHkUmiVkZm9YmabzWxF0L5HzCzTzJYFfi4Kem6oma0zs9Vm1r2sAheRKBI81mDMmKMWuQGYkZ5Jp+EfkzLk33Qa/jEz0jN9CjS2FaWE8CrwD+C1XPtHOueeC95hZqcB/YDmQENgjpmd7Jw7FIJYRSRaFTDWYEZ6JkOnf0P2Qe9jInN7NkOnfwOghW/CrNASgnNuHrC1iNfrBUx2zu13zq0H1gFnlCI+EYl2hYw1ePb91UeSQY7sg4d49v3VYQpQcpSml9EgM/s6UKVUK7AvCdgYdExGYN8xzGyAmS02s8VbtmwpRRgiErGKMNYga3t2nqfmt1/KTkkTwv8CJwFtgE3A3wL78xpZ4vK6gHNutHMu1TmXWq9evRKGISIRq4hjDRom5r3eQX77peyUKCE4535yzh1yzh0GxvBbtVAG0Cjo0GQgq3QhikjUKcZYg8Hdm5EQf/SKaAnxcQzu3qyso5RcSpQQzKxB0K+XATk9kGYB/cysspmlAE2BRaULUUSiSvBYg9mzCx1rkNY2iWG9W5KUmIABSYkJDOvdUg3KPii0l5GZTQLOAeqaWQbwMHCOmbXBqw7aANwC4JxbaWZvAt8CvwK3q4eRSAwp4ViDtLZJSgARoNCE4Jy7Mo/dYws4/kngydIEJSJRKPe6BrnGGkjk01xGIhIaDz6odQ2inBKCiJTeiy/CU0/BzTdrXYMopoQgIqUzbRrccQf07AkvvaR1DaKYEoKIlNz8+XDVVdChA0yeDBU1X2Y0U0IQkZJZsQIuvRSaNIG334bjjvM7IiklJQQRKb6NG6FHD0hIgPfegzp1/I5IQkDlOxEpnm3bvGSwaxfMm+eVEPBmLX32/dVkbc+mYWICg7s309iCKKOEICJFl50NvXrB2rVeyaB1a0BTWJcXqjISkaI5dMhrQJ4/H15/Hc4778hTmsK6fFBCEJHCOQd33glvvQUjR8IVVxz1tKawLh+UEESkcMOGeWMM7rsP7r77mKc1hXX5oIQgIgUbNw7+8hevuujpp/M8RFNYlw9qVBaR/M2e7U1H0bUrvPIKVMj7O2ROw7F6GUU3JQQROSK46+j5u37g5Vfuo2Lr1t70FJUqFXiuprCOfqoyEhHgt66jmduzOWFrJk+PG8qmKjV5d/g/oXp1v8OTMFBCEBHgt66j9XZv47U3vRlL+1/+KE8s2eZzZBIuqjISEcDrIlpt/15emfoIdfdu58p+T7GhdhKmrqMxQyUEEQHgxOPglamPcMqWDQzsNYTlDb0eQuo6GjtUQhAR2L+fSe8+Q53M77ir5318clJ7QF1HY40SgkisO3gQrriC+l/OY+nDfyO9ShtMXUdjkhKCSCw7dAiuvRZmzoQXXqDdoEEs8Dsm8Y3aEERi1eHDcMstMGkSDB8Ogwb5HZH4TAlBJBY5B/fcA2PHwoMPwv33+x2RRAAlBJFY9NBDMGqUN1HdY4/5HY1ECCUEkVgzbBg8+STcdBOMGAFmfkckEUIJQSSWvPACPPAA/OlP8PLLSgZyFPUyEilHClzX+JVXvEVu0tLg1VchLq7Aa0nsUUIQKScKXNd49XyviqhbN5g8GeLj/QxVIpSqjETKifzWNf5i5Djo3x/OOstbArNyZZ8ilEinEoJIOZHX+sWdNizjsamPQru28M47cNxxPkQm0UIlBJFyIvckdKkZKxkz/XE21msE770HNWr4FJlEi0ITgpm9YmabzWxF0L7aZvahma0NPNYKem6oma0zs9Vm1r2sAheRowWva9xy01pemfIoP1Wvy9rXp0Pt2j5HJ9GgKCWEV4EeufYNAT5yzjUFPgr8jpmdBvQDmgfOecnM1JVBJAzS2iYxrHdLumRn8dqbf2V31RqsnvAWF17Qxu/QJEoUmhCcc/OArbl29wLGB7bHA2lB+yc75/Y759YD64AzQhSriBQi7WAm41+/n1p1atDwq8/o0aO93yFJFClpG8LxzrlNAIHH+oH9ScDGoOMyAvuOYWYDzGyxmS3esmVLCcMQkSM+/RTOPx9q1YL58+Gkk/yOSKJMqBuV8xr26PI60Dk32jmX6pxLrVevXojDEIkx770HPXpAcjLMmwcpKX5HJFGopAnhJzNrABB43BzYnwE0CjouGcgqeXgiUqjp0+HSS+GUU7xkkKQFbaRkSpoQZgHXBravBWYG7e9nZpXNLAVoCiwqXYgikq9//Qv++Ec4/XSYOxdU2pZSKHRgmplNAs4B6ppZBvAwMBx408xuBH4ELgdwzq00szeBb4Ffgdudc4fyvLCI5KnA+YiCjR4Nt94K55wDs2ZBtWphj1XKF3Muzyr+sEpNTXWLFy/2OwwR3+Wejwi8he6H9W55dFIYMQLuvRcuvhimTIGEhDyuJuWdmS1xzqWG6noaqSwSQfKbj+jZ91d7vzjnLWhz771w+eVe+4GSgYSI5jISiSB5zUd0ZL9z8D//A889B9ddB//8p6awlpBSCUEkguSejyhHUo3KMHCglwwGDfLWQlYykBBTQhCJIMHzEeWoFgcTP3/ZW+FsyBBvLeQK+q8roacqI5EIktNwnNPLqHG1ikz8ZBRJH8321kF+4AGfI5TyTAlBJMKktU3yEkN2NvTpAx+9C88/D3fd5XdoUs4pIYhEol27vNHHn37qNR7feKPfEUkMUEIQiTQbN0KvXvD11zBhAlx5pd8RSYxQQhCJJIsWeclg7154+2248EK/I5IYoq4KIpFi8mTo0sVb9/iLL5QMJOxUQhApA0Wejwjg8GF49FFvBHLnzt7o47p1wxuwCEoIIiGXez6izO3ZDJ3+DcCxSWHvXm/U8ZQpcP313liDSpXCHLGIR1VGIiFW6HxEOTIz4eyzYepUbwTy2LFKBuIrlRBEQqzA+YhyLFnidSvdudObuvqSS8IUnUj+VEIQCbH85iM6sn/KFK+tID4ePv9cyUAihhKCSIjlNR9RQnwcg7udDI8/7q1w1rat18W0ZUufohQ5lqqMREIs93xEDRMTuL9LYy59drDXtfSaa7zVzipX9jlSkaMpIYiUgSPzEQFs2gRpafDVVzB8uLemgZm/AYrkQQlBpCylp3uNx1u3euML0tL8jkgkX2pDECkr06bBWWd5pYEFC5QMJOIpIYiE2p49cMst0Lev12i8aBG0aeN3VCKFUpWRSD6KNf1EjvR0b3bSNWu8toLHH9dgM4kaSggieSjW9BPgzUc0YoS3olm9ejBnDpx3XjhDFik1VRmJ5KHI00+ANwVFt24weLA3yOzrr5UMJCopIYjkoUjTTwC89Ra0auVNVz1mjNeQXKdOGCIUCT0lBJE8FDr9xJ49MGAA9O4NKSle28FNN2l8gUQ1JQSRPOQ7/UT3Zt7EdO3aeWsd33+/Nx/RySf7FKlI6KhRWSQPeU0/MbhrU9I+nAAPPgj168NHH8G55/ocqUjoKCGI5OOo6ScyMrw5iObOhT59vLmIatf2N0CREFOVkUhhpk/3Go4XLfIWsZkyRclAyiUlBJH8bN0KN97olQhOOslrOL7hBjUcS7lVqiojM9sA7AIOAb8651LNrDbwBtAE2AD80Tm3rXRhipRMiUYbHzrklQQeeAC2bYOhQ+HRR70FbUTKsVCUEM51zrVxzqUGfh8CfOScawp8FPhdJOxyRhtnbs/G8dto4xnpmfmftHAhdOjgzUXUvLlXKnjqKSUDiQllUWXUCxgf2B4PaIpH8UWxRhtv3uxVD3XoAFlZMHEifPKJ13YgEiNKmxAc8IGZLTGzAYF9xzvnNgEEHuvndaKZDTCzxWa2eMuWLaUMQ+RYRRpt/Ouv8MIL3jiC117zJqT77jtvgjq1FUiMKW23007OuSwzqw98aGbfFfVE59xoYDRAamqqK2UcIsdomJhAZh5J4cho43nzYNAg+OYb6NoVRo2CU04Jc5QikaNUJQTnXFbgcTPwFnAG8JOZNQAIPG4ubZAiJZHfaOOH2tWEq6+GLl1gxw6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     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f2(x,r,K):\n",
    "    e = np.exp(r*x)\n",
    "    return (K*P0*e)/(K+P0*(e-1))\n",
    "\n",
    "fit_display(f2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## model 3\n",
    "### formula\n",
    "ODE:\n",
    "$$$\n",
    "\\frac{dP}{dt}=\\gamma(1-\\frac{P}{\\kappa + \\frac{P}{2}})P\n",
    "$$$\n",
    "function:\n",
    "$$$\n",
    "P(t)=2[\\kappa+\\frac{\\kappa-\\frac{P_0}{2}}{P_0 e^{\\gamma t}}(\\kappa-\\frac{P_0}{2}-\\sqrt{2\\kappa P_0 e^{\\gamma t}+(\\kappa-\\frac{P_0}{2})^2})]\n",
    "$$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "parameters: [0.29446449 0.2070968 ]\nloss: 5.56%\n"
    },
    {
     "ename": "ImportError",
     "evalue": "Missing optional dependency 'tabulate'.  Use pip or conda to install tabulate.",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mImportError\u001b[0m                               Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-5-9ca76abe1073>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      6\u001b[0m \u001b[0mresult\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mfit_display\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf3\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 7\u001b[1;33m \u001b[0mresult\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mto_markdown\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'result.md'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      8\u001b[0m \u001b[0mresult\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m~\\AppData\\Roaming\\Python\\Python36\\site-packages\\pandas\\core\\frame.py\u001b[0m in \u001b[0;36mto_markdown\u001b[1;34m(self, buf, mode, **kwargs)\u001b[0m\n\u001b[0;32m   2015\u001b[0m         \u001b[0mkwargs\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msetdefault\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"headers\"\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m\"keys\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2016\u001b[0m         \u001b[0mkwargs\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msetdefault\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"tablefmt\"\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m\"pipe\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2017\u001b[1;33m         \u001b[0mtabulate\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mimport_optional_dependency\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"tabulate\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2018\u001b[0m         \u001b[0mresult\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mtabulate\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtabulate\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2019\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mbuf\u001b[0m \u001b[1;32mis\u001b[0m \u001b[1;32mNone\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m~\\AppData\\Roaming\\Python\\Python36\\site-packages\\pandas\\compat\\_optional.py\u001b[0m in \u001b[0;36mimport_optional_dependency\u001b[1;34m(name, extra, raise_on_missing, on_version)\u001b[0m\n\u001b[0;32m     90\u001b[0m     \u001b[1;32mexcept\u001b[0m \u001b[0mImportError\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     91\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mraise_on_missing\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 92\u001b[1;33m             \u001b[1;32mraise\u001b[0m \u001b[0mImportError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmsg\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mfrom\u001b[0m \u001b[1;32mNone\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     93\u001b[0m         \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     94\u001b[0m             \u001b[1;32mreturn\u001b[0m \u001b[1;32mNone\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mImportError\u001b[0m: Missing optional dependency 'tabulate'.  Use pip or conda to install tabulate."
     ]
    }
   ],
   "source": [
    "def f3(t,r,k):\n",
    "    j = k - P0/2\n",
    "    e = P0*np.exp(r*t)\n",
    "    return 2*(k+j*(j-np.sqrt(2*k*e+j**2))/e)\n",
    "\n",
    "result = fit_display(f3)\n",
    "result.to_markdown('result.md')\n",
    "result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ]
}